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ISSN 1079-6762

 

On spectra of geometric operators on open manifolds and differentiable groupoids


Authors: Robert Lauter and Victor Nistor
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 45-53
MSC (2000): Primary 58J50; Secondary 58H05, 47G30, 58J40
Published electronically: May 8, 2001
MathSciNet review: 1852899
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Abstract:

We use a pseudodifferential calculus on differentiable groupoids to obtain new analytical results on geometric operators on certain noncompact Riemannian manifolds. The first step is to establish that the geometric operators belong to a pseudodifferential calculus on an associated differentiable groupoid. This then leads to Fredholmness criteria for geometric operators on suitable noncompact manifolds, as well as to an inductive procedure to compute their essential spectra. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multicylindrical ends.


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Additional Information

Robert Lauter
Affiliation: Universität Mainz, Fachbereich 17-Mathematik, D-55099 Mainz, Germany
Email: lauter@mathematik.uni-mainz.de

Victor Nistor
Affiliation: Pennsylvania State University, Department of Mathematics, University Park, PA 16802
Email: nistor@math.psu.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-01-00093-2
PII: S 1079-6762(01)00093-2
Keywords: Laplace operator, pseudodifferential operator, $C^*$-algebra, groupoid, essential spectrum
Received by editor(s): May 30, 2000
Published electronically: May 8, 2001
Additional Notes: Lauter was partly supported by a scholarship of the German Academic Exchange Service (DAAD) within the Hochschulsonderprogramm III von Bund und Ländern, and the Sonderforschungsbereich 478 Geometrische Strukturen in der Mathematik at the University of Münster. Nistor was partially supported by NSF Young Investigator Award DMS-9457859 and NSF Grant DMS-9971951.
Communicated by: Michael Taylor
Article copyright: © Copyright 2001 American Mathematical Society